The noise started after I got a flat tire on the road. A description of everything I know is below.

1) I get a thumping noise that sounds like its coming from the rear end.
2) I get it in a straight line and when I corner (although cornering seems to make it happen less often)
3) It happens about 80-90% of the time after I have stopped and begin moving again
4) It builds in intensity from stop up to about 20mph then abruptly stops
4.5) It has never made any noise above 25mph - no matter what I do.
5) It seems to increase frequency with vehicle speed but I can't tell if its at the wheel or the driveshaft frequency...
6) if I just try to gas it through the noise, it gets sufficiently loud to make me worry about the back end falling off and I end up letting off the gas
7) If I let off the gas when I start to hear it or (very) gingerly accelerate through, it can be made very quiet.
8) It happens in any gear, so long as I'm going slowly (I tried it in 1st, 2nd, 3rd)
9) I don't have to stop to hear it, if I slow down below 20 it happens upon accelerating again

I originally thought it was a CV joint or something, but I figured that problem would occur at all speeds and I see no evidence of boot wear.

Next I thought it might be the driveshaft U Joint or something like that but its hard to inspect. I need to try harder.

I checked the RTAB's and ensured that they were ok and not the source of the noise. (They are upgrade bushings I installed last year, and they aren't the cheapo ones) The car's alignment still seems ok.

I'm getting ready to replace the diff fluid and remove the back cover to inspect the inners...

The noise started after I got a flat tire on the road. A description of everything I know is below.

1) I get a thumping noise that sounds like its coming from the rear end.
2) I get it in a straight line and when I corner (although cornering seems to make it happen less often)
3) It happens about 80-90% of the time after I have stopped and begin moving again
4) It builds in intensity from stop up to about 20mph then abruptly stops
4.5) It has never made any noise above 25mph - no matter what I do.
5) It seems to increase frequency with vehicle speed but I can't tell if its at the wheel or the driveshaft frequency...
6) if I just try to gas it through the noise, it gets sufficiently loud to make me worry about the back end falling off and I end up letting off the gas
7) If I let off the gas when I start to hear it or (very) gingerly accelerate through, it can be made very quiet.
8) It happens in any gear, so long as I'm going slowly (I tried it in 1st, 2nd, 3rd)
9) I don't have to stop to hear it, if I slow down below 20 it happens upon accelerating again

I originally thought it was a CV joint or something, but I figured that problem would occur at all speeds and I see no evidence of boot wear.

Next I thought it might be the driveshaft U Joint or something like that but its hard to inspect. I need to try harder.

I checked the RTAB's and ensured that they were ok and not the source of the noise. (They are upgrade bushings I installed last year, and they aren't the cheapo ones) The car's alignment still seems ok.

I'm getting ready to replace the diff fluid and remove the back cover to inspect the inners...

Any suggestions are welcome

The last time I had a flat I also experienced a noise in the rear of the car. The noise eventually got louder with time.

It turned out it was a shock mount that got damaged while driving with the run flat. Replaced it as well as the rear shocks (somehow got worn by the overwork) and all is quite.